special values

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31—40 of 113 matching pages

31: 28.12 Definitions and Basic Properties

… ►(28.12.10) is not valid for cuts on the real axis in the

q

-plane for special complex values of

\nu

; but it remains valid for small

q

; compare §28.7. …

33: 25.2 Definition and Expansions

… ►

25.2.2

\zeta\left(s\right)=\frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}},

\Re s>1

ⓘ

Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\Re$ : real part, $n$ : nonnegative integer and $s$ : complex variable
Keywords:: infinite series
Proof sketch:: Derivable from (25.2.1).
A&S Ref:: 23.2.20 (is the special case with integer values of $s$ )
Referenced by:: (25.11.11)
Permalink:: http://dlmf.nist.gov/25.2.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §25.2(ii), §25.2 and Ch.25

►

25.2.3

\zeta\left(s\right)=\frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^% {s}},

\Re s>0

ⓘ

Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\Re$ : real part, $n$ : nonnegative integer and $s$ : complex variable
Keywords:: infinite series
Source:: Apostol (1976, (27), p. 292)
A&S Ref:: 23.2.19 (is the special case with integer values of $s$ )
Referenced by:: §25.11(x)
Permalink:: http://dlmf.nist.gov/25.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.2(ii), §25.2 and Ch.25

…

34: 13.4 Integral Representations

… ►The fractional powers are continuous and assume their principal values at

t=\alpha

. …At the point where the contour crosses the interval

(1,\infty)

t^{-b}

and the

{{}_{2}{\mathbf{F}}_{1}}

function assume their principal values; compare §§15.1 and 15.2(i). A special case is … ►Again,

t^{-c}

and the

{{}_{2}{\mathbf{F}}_{1}}

function assume their principal values where the contour intersects the positive real axis. …

35: Bille C. Carlson

… ►Also, the homogeneity of the

R

-function has led to a new type of mean value for several variables, accompanied by various inequalities. ►The foregoing matters are discussed in Carlson’s book Special Functions of Applied Mathematics, published by Academic Press in 1977. …

36: Mathematical Introduction

… ►With two real variables, special functions are depicted as 3D surfaces, with vertical height corresponding to the value of the function, and coloring added to emphasize the 3D nature. … ►Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. …

37: 16.4 Argument Unity

… ►The special case

{{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};1\right)

is $k$ -balanced if

a_{q+1}

is a nonpositive integer and … ►The function

{{}_{q+1}F_{q}}

with argument unity and general values of the parameters is discussed in Bühring (1992). Special cases are as follows: … ►These series contain

\mathit{6j}

symbols as special cases when the parameters are integers; compare §34.4. …

38: 18.9 Recurrence Relations and Derivatives

… ►with initial values

p_{0}(x)=1

and

p_{1}(x)=A_{0}x+B_{0}

. … ►with initial values

p_{0}(x)=1

and

p_{1}(x)=a_{0}^{-1}(x-b_{0})

. … ►Formulas (18.9.5), (18.9.11), (18.9.13) are special cases of (18.2.16). Formulas (18.9.6), (18.9.12), (18.9.14) are special cases of (18.2.17). …

39: 23.5 Special Lattices

§23.5 Special Lattices

►

§23.5(i) Real-Valued Functions

►The Weierstrass functions take real values on the real axis iff the lattice is fixed under complex conjugation:

\mathbb{L}=\overline{\mathbb{L}}

; equivalently, when

g_{2},g_{3}\in\mathbb{R}

. …

40: 35.1 Special Notation

§35.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►All fractional or complex powers are principal values. ► ►►►

$a, b$	complex variables.
…
$f(\mathbf{X})$	complex-valued function with $\mathbf{X}\in{\boldsymbol{\Omega}}$ .
…

►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively

\Gamma_{m}\left(a\right)

and

\mathrm{B}_{m}\left(a,b\right)

, and the special functions of matrix argument: Bessel (of the first kind)

A_{\nu}\left(\mathbf{T}\right)

and (of the second kind)

B_{\nu}\left(\mathbf{T}\right)

; confluent hypergeometric (of the first kind)

{{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)

\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)